Time-state Claims

Contents:

An important subfield of physics -- nuclear physics -- deals
with the smallest particles of which matter is composed.
Constructs developed by Kenneth Arrow [Arrow
1964] and Gerard Debreu [Debreu
1959] provided a similar foundation for financial economics.
The resulting approach is often called the Arrow-Debreu
Paradigm. It characterizes promised future payments in terms
of both the times at which payments are to be made and
the states of the world that must obtain for payments to
be made. Hence the often-used name: the time-state paradigm.

Since the approach represents securities and other types of
financial instruments in terms of their most elemental
components, one could as well title it: Nuclear Financial
Economics.

We start with the simplest possible example that involves both
time and uncertainty. There are two time
periods:

0: today
1: a year from now

There are also two possible future states of the world:

G: the weather over the next year will be good
B: the weather over the next year will be bad

The states of the world are mutually exclusive (if
one occurs, the other cannot) and exhaustive (one of
them must occur).

The economy is very simple indeed. The only commodity is the apple
and there is no money per se. In effect, the apple is
the unit of currency.

The only type of productive investment in this economy is, not
surprisingly, an apple tree. We focus initially on a
tree that will produce:

63 apples if the weather is good
48 apples if the weather is bad

This is shown in the figure below. Time proceeds from left to
right. The box on the left refers to present value. The
boxes on the right represent alternative states of the world. One
but only one of the states of the world in a vertical position
will take place at the time in question. The names of the states
are indicated at the tops of the boxes. The numbers inside the
boxes indicate the payoffs.

One apple at time 0 (today)
One apple at time 1 if the weather is good
One apple at time 1 if the weather is bad

In keeping with our interpretation of the Arrow-Debreu
approach as Nuclear Financial Economics, we will call these atomic
time-state claims. Note that the latter two descriptions include
the item (apple), the number of units (one), the time at which
delivery is to be made (0 or 1), and the state of the world that
must obtain for delivery to be made (good or bad weather). In the
case of the first description, no state is given, since present
values are not conditional on future states of the world. To
simplify the exposition, we will refer to these claims as:

Many of the fundamental concepts of Financial Economics are
based on the assumption that markets exist in which
claims can be traded efficiently (at low cost). We begin with the
assumption that dealers stand ready to trade atomic
claims and to do so without cost. As will be seen, these dealers
are a bit of an artifice. Later we consider more realistic
assumptions about the world.

Assume that Dealer G "makes a market" in
good weather apples. In particular, she is willing to trade
(swap):

This can better be understood in standard financial terms.
Assume that an owner of an apple tree has issued a certificate of
the following form:

I, __________ promise to deliver to the bearer of this certificate
one apple at the end of year ____ if (but only if) the weather
during the year has been good.

In standard parlance, this piece of paper (or its electronic
equivalent) would be termed a security. Assume that a credit-rating
agency has examined the property of the apple grower (the
apple tree) and has established that no more than 63 of these
securities have been issued and that there are no other claims on
the grower's assets in the event of good weather. As a result,
the securities are rated AAA ("triple-A") and
can be considered default-free.

Under these conditions, the security in question represents a
property right in an atomic time-state claim. In a sense, it is
thus an atomic security or, given the origin of the
concept, an Arrow-Debreu security.

The price of this security is 0.285 present apples,
since the dealer stands ready to trade this number of present
apples for the security. More generally, the price of any
security is the amount of the relevant numeraire paid immediately
for which the security can be traded. Note that the ability to
make a trade is central to the definition of a price.

In the real world, of course, dealers charge more to sell a
security than they are willing to pay to buy it. The spread
between the ask (selling) and bid (buying)
price provides compensation for the market-making function. In
our examples we assume (unrealistically) that there is no such
spread and hence that there is but one price. In practice, the
average of the bid and ask prices is often used as a surrogate
for "the price". For detailed computations, of course,
the specifics of a proposed transaction may need to be taken into
account and the relevant price (bid or ask) used.

This diversion completed, we return to our world of non-profit
dealers.

In addition to Dealer G, we assume that another, Dealer B, is
willing to trade (swap):

Thus far we have a world with three types of time-state claims
(PA, GA and BA). Explicit markets exist for trading (1) PA and GA
and (2) PA and BA. Note that each such trade has the
characteristic of an investment -- today's goods are
traded for the prospect of goods in the future. Thus one
purchasing a GA atomic security can be said to have invested
0.285 (present) apples to obtain 1.000 apples in the future if
the weather is good.

But what of the other possible type of trade in this world?
What would it mean to trade good weather apples for bad weather
apples? How might one accomplish this? And what would be the
terms of trade?

To answer these questions, consider the following agreement:

Party A promises to pay party B: 6 apples if the weather is good
Party B promises to pay party A: 3 apples if the weather is bad
Neither party pays the other anything today (on signing)

Such an agreement is called a swap in financial
parlance. It represents the third possible type of trade in our
simple world: GA for BA.

Is this a fair deal? If one desires an answer based
on ethical considerations, other disciplines will have to be
invoked. Financial Economics can only indicate whether or not one
of the parties could get a better deal elsewhere.

Assume that Party A comes to you with the proposal that you
sign the agreement as Party B. You are willing to give up 3
apples if the weather is bad in order to increase your
consumption if the weather is good. But is 6 apples the best that
you can do?

The net result is, of course, to trade 3 BA for 7 GA -- a
better deal than offered by canny Party A, who will have to
search elsewhere for a counterparty foolish enough to
take the deal.

Note that although explicit markets are being made in only
future atomic time-state claims, it is possible to
"create" trades involving any present and future
claims. This is a perfectly general result. If one can trade each
possible future atomic time-state claim for present units of a
numeraire, any desired trade can be accomplished. Thus a set of
atomic security prices is sufficient for accomplishing
any desired trade.

Consider two people sharing a pizza. To insure an even
division, it is wise to agree that one party should cut it, and
the other should choose his or her piece. Similarly, it is useful
to require someone offering a bet on a sporting event to be
willing to take either side on the offered terms. With this in
mind, we return to Party A and Party B.

Assume that a securities firm is willing to serve as either
Party A or Party B in the previously-described swap (6 GAs for 3
BAs). Clearly, you have no interest in being Party B. But what
about serving as Party A? Consider the following set of trades:

It is useful to put all this information in a payment
matrix with each row representing a time-state combination
and each column a transaction. Conventionally, we represent
outflows with negative numbers, inflows with positive numbers,
and neither with zeros.

Note what this set of transactions accomplishes -- getting
something for nothing! Moreover, there is no reason to settle for
such a small gain. Double the sizes of all the transactions and
the net gain is doubled. Quadruple them and the gain is
quadrupled.

Well and good, but what if one really wanted apples next year
if the weather is good. Not to worry. Add a final trade in which
0.285 present apples are traded for 1.000 good weather apples.
Want bad weather apples? Add a trade to convert the gains into
the appropriate payment. No matter what a person's preferences
may be, it is desirable to exploit the foolishness of the firm
offering this swap.

Too good to be true? Probably. This example constitutes an
arbitrage -- every trader's dream. To formalize:

An arbitrage provides a positive net payoff in at least one time
and state and no negative net payoff in any time and state.

An arbitrage is thus a money machine (or, as in this case, an
apple machine). When an opportunity of this type arises, traders
will rush to exploit it, causing others to adjust their terms of
trade until swap terms involve no arbitrage.

In an important sense, every security transaction can be
considered a swap. The purchase of an atomic security is
a swap of present goods for conditional future goods. The sale of
such a security is a swap of conditional future goods for present
goods. Such cases, when one "side" of the swap involves
present goods or services, are typically termed investments.
Thus one invests present apples in the hope of obtaining
more apples in the future. But note that the swap of good weather
apples for bad weather apples is no different in kind, even
though no goods or services are exchanged at the time of the
agreement.

To be explicit, we refer to swaps of this latter kind as zero-investment
strategies. As with other transactions, they are represented
by cash flow vectors with positive and negative numbers, but with
zeros in the first (present) row.